Fourier–Jacobi coefficients of Eisenstein series on unitary groups

Abstract

This paper studies the Fourier–Jacobi expansions of Eisenstein series on $U\left(3,1\right)$. I relate the Fourier–Jacobi coefficients of the Eisenstein series with special values of $L$-functions. This relationship can be applied to verify the existence of certain Eisenstein series on $U\left(3,1\right)$ that do not vanish modulo $p$. This is a crucial step towards one divisibility of the main conjecture for ${GL}_2\times K^{\times }$ using the method of Eisenstein congruences.